
Then, by partitioning the set of random variables \(X_{T_l^{\prime }}\)  into \(k_{\rm {ps},l}\)  parts and using the same arguments as [1]} with \(f_1=f_2=\cdots =f_n=f\) , we can show that
\(\mathbb {E}\big [e^{\theta \sum _{i \in \mathcal {T}^{\prime }_l} f(X_i)}\big ]&\le \exp \bigg (\frac{2(|\mathcal {T}^{\prime }_l|+1/\gamma _{\rm {aps}})\operatorname{\mathsf {Var}}_{\pi }(f)}{\gamma _{\rm {ps,l}}}\theta ^2 \bigg (1-\frac{10\theta }{\gamma _{\rm {ps},l}}\bigg )^{-1}\bigg ) \\&\le \exp \bigg (\frac{2(|\mathcal {T}^{\prime }_l|+1/\gamma _{\rm {aps}})|\operatorname{\mathsf {Var}}_{\pi }(f)}{\gamma _{\rm {aps}}}\theta ^2 \bigg (1-\frac{10\theta }{\gamma _{\rm {aps}}}\bigg )^{-1}\bigg ) ,\) 
